# Unit 6 for R
# 030528 prepared by C. Andy Tsao
# Topic : Simple Linear Regression
# cf. Chapter 11 of Johnson and Bhattacharyya
# We will use the data in Table 1 (cf. also Table 4) P 480,489
# to illustrate the steps, ideas and output of simple 
# linear regression

allergy<-matrix(scan("C11T1.dat"),ncol=2,byrow=T)
dose<-allergy[,1];dura<-allergy[,2];

# prelim analysis
# Initiate graphic devices
par(mfrow=c(2,2));
stem(dose);stem(dura);
hist(dose);hist(dura);
boxplot(dose);boxplot(dura)

# Prelim Normal Check
 hist(dura); boxplot(dura); plot(density(dura));
 qqnorm(dura); qqline(dura);
 
# Plot "y" vs "x"
plot(dura~dose)

# Simple Linear Regression
fm0<-lm(dura~1)  # with intercept only
summary(fm0)

fm1<-lm(dura~dose)  # Y = b_0 + b_1 X + epsilon
summary(fm1)

# Plot the fitted line overlayed over the y vs. x scatter plot
plot(dose,dura,xlab="dose", ylab="dura", main="Simple Linear Regression");
abline(coef(fm1))

# Residual analysis: Outliers and influential points; Normality

r<-fm1$res
par(mfrow=c(2,2));
hist(r); boxplot(r); plot(density(r));
 qqnorm(r); qqline(r);
 
# An experiment of effects of outliers
dura[1]
dura[1]<-90
# Now redo the analysis and compare with previous output